The base five numeral system, also known as the quinary place system, could be taught using a whole range of activities using pieces of equipment that most of your students probably already have - the set of five fingers on one of their hands.
Begin by explaining the method of counting and recording the count in the quinary system. If using tally marks, for example, the students would make one mark for each of the four fingers, cross over the four marks when counting the thumb, and then start another set of tally marks. That count would be read as "one-zero base five" since there is no "ten" in this counting system.
Once the students understand how to count and how to read numerals in base five notation, you could present whatever activities, story problems, or other exercises you wished, depending upon the level and abilities of your students.